The Farrell-Jones Conjecture for mapping class groups

TL;DR

This paper proves the Farrell-Jones Conjecture for mapping class groups by leveraging their large-scale geometry and the properties of Teichmueller space, extending existing axiomatic frameworks.

Contribution

It extends the Farrell-Jones Conjecture to mapping class groups using an axiomatic approach based on the geometry of Teichmueller space.

Findings

Proof of the Farrell-Jones Conjecture for mapping class groups

Extension of projection axioms in geometric group theory

Demonstration of finite F-amenability of the group action

Abstract

We prove the Farrell-Jones Conjecture for mapping class groups. The proof uses the Masur-Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmueller space. The proof is presented in an axiomatic setup, extending the projection axioms of Bestvina-Bromberg-Fujiwara. More specifically, we prove that the action of the mapping class group on the Thurston compactification of Teichmueller space is finitely F-amenable for the family F consisting of virtual point stabilizers.